Question

find the equation of tangent to the circle x²+y²=a² making area of triangle a² with the axis

Answer 1

See the diagram.

Take a tangent to circle at P, which intercepts X axis at C(c,0) and y axis at D(0,d). Its equation is given by :  c y + d x - cd = 0        It is simple that.

Area of triangle DOC is = $\frac{1}{2} c d = a^2 \ \ \ \ \ --\ equation\ 1$

CPD is a tangent and its distance from origin is the radius = a.
Distance from O(0,0) to line cy+dx - cd = 0 is given by

$\frac{| c 0 + d 0 - c d |}{\sqrt(c^2+d^2)}} = a \\ \\ \\ \frac{c^2 d^2}{c^2 + d^2} = a^2 \ \ \ -- equation\ 2 \\ \\ Comparing\ equation\ 1\ and\ 2,\ we\ get\ \\ \\ c^2 + d^2 = 2 c d \\ \\ So,\ c = d \\ \\ From\ equation\ 1,\ we\ get c = d = \sqrt{2} a \\ \\ Substituting\ in\ \equation\of\ tangent,\we\ get\ y + x = \sqrt{2}a$